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Bergman Kernel Function for Hartogs Domains Over Bounded Homogeneous Domains
Ishi, H.,Park, J. D.,Yamamori, A. MATHEMATICA JOSEPHINA, INC. 2017 JOURNAL OF GEOMETRIC ANALYSIS Vol.27 No.2
<P>We obtain an explicit formula of the Bergman kernel for Hartogs domains over bounded homogeneous domains. In order to find a simple formula, we consider a Siegel domain biholomorphic to the bounded homogeneous domain and use its Bergman kernel obtained by Gindikin. The Bergman kernel of the Hartogs domain is expressed by two different forms and the main part of the Bergman kernel is a polynomial whose coefficients contain the Stirling number of the second kind. As an application of our formula, we investigate the Lu Qi-Keng problem for our Hartogs domains and give some important examples of Hartogs domains whose Bergman kernels are zero-free.</P>
On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature
MATHEMATICA JOSEPHINA, INC. 2017 JOURNAL OF GEOMETRIC ANALYSIS Vol.27 No.2
<P>We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonicWeyl curvature, i.e., delta W = 0. Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product R-2 x N. of the Euclidean metric and a 2-d Riemannian manifold of constant curvature. lambda not equal 0, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works (in Trans Am Math Soc 364: 2377-2391, 2012; DukeMath J 162: 10031204, 2013) and Derdzinski's study on Codazzi tensors (in Math Z 172: 273-280, 1980). Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with delta W = 0. For the shrinking case, it re-proves the rigidity result (Fernandez-Lopez and GarciaRio inMath Z 269: 461- 466, 2011; Munteanu and Sesum in J. Geom Anal 23: 539- 561, 2013) in 4- d. It also helps to understand the expanding case; we now understand all 4- d non- conformally flat ones with delta W = 0. We also characterize locally 4- d (not necessarily complete) gradient Ricci solitons with harmonic curvature.</P>
Generalizations of the Choe–Hoppe Helicoid and Clifford Cones in Euclidean Space
Lee, E.,Lee, H. MATHEMATICA JOSEPHINA, INC. 2017 JOURNAL OF GEOMETRIC ANALYSIS Vol.27 No.1
<P>By sweeping out L independent Clifford cones in R2N+2 via the multi-screw motion, we construct minimal submanifolds in RL(2N+2)+1. Also, we sweep out the L-rays Clifford cone (introduced in Sect. 2.3) in RL(2N+2) to construct minimal submanifolds in RL(2N+2)+1. Our minimal submanifolds unify various interesting examples: Choe-Hoppe's helicoid of codimension one, the cone over Lawson's ruled minimal surfaces in S-3, Barbosa-Dajczer-Jorge's ruled submanifolds, and Harvey-Lawson's volume-minimizing twisted normal cone over the Clifford torus 1/root 2S(N) x 1/root 2S(N).</P>